Theorem mprg 2420

Description: Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)

Hypotheses

Ref	Expression
mprg.1	|- ( A. x e. A ph -> ps )
mprg.2	|- ( x e. A -> ph )

Assertion

Ref	Expression
mprg	|- ps

Proof of Theorem mprg

Step	Hyp	Ref	Expression
1		mprg.2	. . 3 |- ( x e. A -> ph )
2	1	rgen 2416	. 2 |- A. x e. A ph
3		mprg.1	. 2 |- ( A. x e. A ph -> ps )
4	2, 3	ax-mp 7	1 |- ps

Syntax hints:   -> wi 4   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by:  reximia  2456  rmoimia  2792  iuneq2i  3696  iineq2i  3697  dfiun2  3712  dfiin2  3713  dfiun3  4609  dfiin3  4610  cnviinm  4879  bj-omtrans  10751